I occasionally experience a blockage of my black tank when draining. We use "normal" house toilet paper for a myriad of reasons. Yes, I know it's not recommended. But I found a quick and easy way to clear a blocked line. I use the gray water tank, it should be full or nearly so to accomplish this.
I purchased an accessory gate valve which connects to the plumbing drain line and then connect my dump pipe to that. This allows me shut off the dump flow while leaving the gray/black valves open. Don't panic yet, I'm not contaminating grey with black. If I see I've got a black blockage I close all valves, wait until the flow stops and then close the accessory valve. Then I open the grey water tank until the pipes are full, usually gurgling followed by silence. Then I open the black water valve. The pressure of the gray water is greater than the black and the gray water backflows into the black tank. I wait a about 30 seconds, then close the gray water valve and then open the accessory valve.
This works great for me, never failed to clear a black blockage in seconds.
2012 Ram 2500 4x4 Lariat Longhorn 6.7 CTD HO, Edge Evo CTS, Extreme Tow/Haul brakes, aFePower Diff cover, LL 5000 bags, 285/70R17, Reese Q20 w/slider. 2005 Montana 2955RL w/400w solar, Renogy MPPT, 4 x 6v @ 12VDC (450AH), 3000w Inverter, King VQ4100.